... One of my neighbors... said he gave up keeping birds because he was tired of losing them (they even got his large ducks.) So, now I face the thought of never being able to have them again....
When discussing varmint proofing a chicken coop, house, pen, or run, you need to use an "integrated" form of pest management. Integrated pest management is taught at all the top Agricultural Universities.
Integrated pest management dose nothing until a farmer's losses reach a predetermined level of economic loss. The next step is building a better chicken pen or coop and when the losses are severe enough Integrated pest management requires you to kill without compunction, compassion, or remorse every living organism that is responsible for your loses.
No one has a herd of mink, they are incredibly unsocial animals who naturally fight and kill one another. With some hard work and an investment in building materials you can reduce your mink losses to one or two raids a year, maybe less. If that number of Mink kills is too high for you then a serious varmint dog is in order along with a few mink traps to thin the mink population to slightly below the natural carrying capacity of the land.
Google building a mink box.
A somewhat similar contraption also does a swell job catching weasels, or the minks' little brother.